Spherical Geometry - Key Properties

Spherical geometry's positive curvature creates unique properties fundamentally different from Euclidean and hyperbolic cases.

DefinitionAntipodal Points

Every point $p$ on a sphere has an antipodal point $-p$ (diametrically opposite). All great circles through $p$ also pass through $-p$ . The distance between antipodal points is $\pi R$ (half the sphere's circumference).

Unlike Euclidean geometry, two great circles always intersect—at exactly two antipodal points. This means there are no parallel lines in spherical geometry, violating the Euclidean parallel postulate in the opposite direction from hyperbolic geometry.

ExampleSpherical Trigonometry Laws

For a spherical triangle with sides $a, b, c$ and opposite angles $A, B, C$ :

Spherical law of cosines:

\cos c = \cos a \cos b + \sin a \sin b \cos C

Spherical law of sines:

\frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin C}{\sin c}

These reduce to Euclidean formulas for small triangles where $\sin x \approx x$ and $\cos x \approx 1 - x^2/2$ .

Lunes (regions between two great circles) have area proportional to their angle. A lune with angle $\theta$ (in radians) has area $2R^2\theta$ . This follows from the sphere's rotational symmetry.

Remark

The circumference of a circle of spherical radius $r$ is $2\pi R \sin(r/R)$ , which is less than $2\pi r$ (the Euclidean value). As curvature increases, circles "shrink"—another manifestation of positive curvature.

Triangles with three right angles exist in spherical geometry (e.g., on Earth's surface, the triangle formed by the equator and two meridians 90° apart). Such triangles have area $\pi R^2/2$ (one-eighth of the sphere).